This type of \(\mathrm {GR}\)  modification is endowed with a rich
cosmological background and may be utilized to explore a sequence of
cosmic events (i.e., primordial inflation, matter-dominated stage,
transition of deceleration to cosmic acceleration stage and current
cosmic speed-up, etc.)
[1]}, [2]}, [3]}. In
addition, the concerns of \(\mathrm {DE}\)  as well as \(\mathrm {DM}\)  may
also be figured out via this modification of gravity. In the
atmosphere of well-known \(f(\mathcal {G})\)  model, Myrkzakulov
et al. [4]} explored the
cosmological solutions in particular \(\Lambda \)\(\mathrm {CDM}\)  model.
They also demonstrated that such type of theory may address the DE
contributions as well as the period of inflation. Within the limits
of string-motivated \(f(\mathcal {G})\)  cosmology, Odintsov and
Oikonomou [5]} investigated the gravitational
baryogenesis by formulating a coupling between baryonic current and
the scalar \(\mathcal {G}\) . Oikonomou [6]}
described different cosmic evolution phases with the assistance of
\(\mathbf {R}+f(\mathcal {G})\)  \(\mathrm {DE}\)  models. Felice and
Tsujikawa [3]} discussed the cosmological
evolution for various \(f(\mathcal {G})\)  models and explained certain
conditions for the viability of these explicit models.
